
Observe that \(\bar{\Pi }\)  is non-negative and primitive, therefore, by the Perron-Frobenius Theorem (see [1]}), there is a unique positive eigenvalue \(\lambda _1\) , that is strictly larger in modulus than any other eigenvalue, and corresponding strictly positive left and right eigenvectors
\(\mathbf {u}\)  and \(\mathbf {v}\) , such that \(\bar{\Pi }^n=\frac{\lambda _1^n}{\mathbf {u}\mathbf {v}}\mathbf {v}\mathbf {u}+\mathcal {O}(n^{|J|}|\lambda _2|^n)\)  element-wise,
where \(\lambda _2\)  is an eigenvalue of second largest modulus. This leads to
\(\mathbb {P}(X_{k_i,m_i}^{(n)}=0,i\in I)=\frac{(\pi \mathbf {v})(\mathbf {u}{1})}{\mathbf {u}\mathbf {v}}\lambda _1^n+\mathcal {O}(n^{|J|}|\lambda _2|^n).\) 
